= Two-level sudden quench
{title2=$p_g=\cos^2(\theta/2),\quad p_e=\sin^2(\theta/2)$}
Take an isolated <two-level system>, or a <spin one-half> <qubit>, with <Pauli matrices> $\sigma_x,\sigma_z$. Let $\sigma_z|0\rangle=|0\rangle$ and $\sigma_z|1\rangle=-|1\rangle$. The initial and final <Hamiltonian operators> are
$$
H_i=-\frac{\Delta_i}{2}\sigma_z,\qquad
H_f=-\frac{\Delta_f}{2}(\cos\theta\,\sigma_z+\sin\theta\,\sigma_x),\qquad \Delta_i,\Delta_f>0.
$$
The <energy> gaps are $\Delta_i,\Delta_f$. Prepare the initial <ground state> $|0\rangle$, with <energy eigenvalue> $-\Delta_i/2$. A final <orthonormal eigenbasis> is
$$
|g_f\rangle=c|0\rangle+s|1\rangle,\qquad
|e_f\rangle=-s|0\rangle+c|1\rangle,\qquad
c=\cos(\theta/2),\quad s=\sin(\theta/2).
$$
Direct <matrix multiplication> gives $H_f|g_f\rangle=-\Delta_f|g_f\rangle/2$ and $H_f|e_f\rangle=+\Delta_f|e_f\rangle/2$. The unchanged initial <quantum state> is $|0\rangle=c|g_f\rangle-s|e_f\rangle$. Its new <probability amplitudes> are the <inner products> $c,-s$; the <Born rule> gives $p_g=c^2,p_e=s^2$. The angle between the real amplitude vectors is $\theta/2$, whereas the Hamiltonian axis on the <Bloch sphere> rotates through $\theta$.
\Image[/sudden-quench-eigenbasis-overlaps.png]
{title=Fixed <quantum state>, rotated <eigenbasis>, and sudden-quench <Born rule> <probabilities>}
{description=Original <two-level sudden quench> calculation. The left panel shows real amplitude coordinates, not spatial directions or the <Bloch sphere>. Dashed perpendicular projections give signed <probability amplitudes>; the right panel squares them. At $\theta=\pi/3$, the final <ground state> and excited-state <probabilities> are $3/4$ and $1/4$.}
{source=sudden-quench-eigenbasis-overlaps.py}
At $\theta=\pi/3$, $c=\sqrt3/2$ and $s=1/2$, so the final <energy measurement> gives $-\Delta_f/2$ with <probability> $3/4$ and $+\Delta_f/2$ with <probability> $1/4$. At $\theta=0$ the initial <ground state> remains the final <ground state>; at $\theta=\pi$ it is the final excited <energy eigenstate>. These statements are about the new <energy> labels, not an instantaneous physical rotation of the <quantum state>.
Back to article page
